5. Which of the following statements are true about elastic and inelastic collisions?

Perfectly elastic and perfectly inelastic collisions are the two opposite extremes along a continuum; where a particular collision lies along the continuum is dependent upon the amount kinetic energy which is conserved by the two objects.

Most collisions tend to be partially to completely elastic.

Momentum is conserved in an elastic collision but not in an inelastic collision.

The kinetic energy of an object remains constant during an elastic collision.

Elastic collisions occur when the collision force is a non-contact force.

Most collisions are not inelastic because the collision forces cause energy of motion to be transformed into sound, light and thermal energy (to name a few).

A ball is dropped from rest and collides with the ground. The higher that the ball rises upon collision with the ground, the more elastic that the collision is.

A moving air track glider collides with a second stationary glider of identical mass. The first glider loses all of its kinetic energy during the collision as the second glider is set in motion with the same original speed as the first glider. Since the first glider lost all of its kinetic energy, this is a perfectly inelastic collision.

The collision between a tennis ball and a tennis racket tends to be more elastic in nature than a collision between a halfback and linebacker in football.

11. In order to catch a ball, a baseball player naturally moves his or her hand backward in the direction of the ball's motion once the ball contacts the hand. This habit causes the force of impact on the players hand to be reduced in size principally because ___.

18. In a physics experiment, two equal-mass carts roll towards each other on a level, low-friction track. One cart rolls rightward at 2 m/s and the other cart rolls leftward at 1 m/s. After the carts collide, they couple (attach together) and roll together with a speed of _____________. Ignore resistive forces.

21. Two objects, A and B, have the same size and shape. Object A is twice as massive as B. The objects are simultaneously dropped from a high window on a tall building. (Neglect the effect air resistance.) The objects will reach the ground at the same time but object A will have a greater ___. Choose all that apply.

24. A wad of chewed bubble gum is moving with 1 unit of momentum when it collides with a heavy box that is initially at rest. The gum sticks to the box and both are set in motion with a combined momentum that is ___.

26. Consider the concepts of work and energy (presuming you have already studied it) and those of impuse and momentum. Force and time is related to momentum change in the same manner as force and displacement pertains to ___________.

30. When a mass M experiences a velocity change of v in a time of t, it experiences a force of F. Assuming the same velocity change of v, the force experienced by a mass of 2M in a time of (1/2)t is ____.

31. When a mass M experiences a velocity change of v in a time of t, it experiences a force of F. Assuming the same velocity change of v, the force experienced by a mass of 2M in a time of (1/4)t is ____.

32. When a mass M experiences a velocity change of v in a time of t, it experiences a force of F. Assuming the same velocity change of v, the force experienced by a mass of (1/2)M in a time of (1/2)t is ____.

33. When a mass M experiences a velocity change of v in a time of t, it experiences a force of F. Assuming the same velocity change of v, the force experienced by a mass of (1/2)M in a time of 4t is ____.

34. A 0.5-kg ball moving at 5 m/s strikes a wall and rebounds in the opposite direction with a speed of 2 m/s. If the impulse occurs for a time duration of 0.01 s, then the average force (magnitude only) acting upon the ball is ____ Newtons.

Part C: Diagramming and Analysis

For Questions #37-#40: Consider the before- and after-collision momentum vectors in the diagram below. Determine the magnitude and direction of the system momentum before and after the collision and identify whether or not momentum is conserved. Finally, determine the magnitude and direction of the net external impulse encountered by the system during the collision.

For Questions #50-#52, determine the total kinetic energy of the system before and after the collision and identify the collision as being either perfectly elastic, partially inelastic/elastic or perfectly inelastic.

Part D: Qualitative Relationships Between Variables

53. An object with a mass M and a velocity v has a momentum of 32 kg•m/s. An object with a mass of ...

.. 2M and a velocity of 2v would have a momentum of 64 kg•m/s.

... 2M and a velocity of 0.5v would have a momentum of 32 kg•m/s.

... 0.5M and a velocity of 2v would have a momentum of 32 kg•m/s.

... 0.5M and a velocity of 0.5v would have a momentum of 8 kg•m/s.

... 4M and a velocity of v would have a momentum of 128 kg•m/s.

... 4M and a velocity of 0.5v would have a momentum of 64 kg•m/s.

... 0.5M and a velocity of 4v would have a momentum of 64 kg•m/s.

... 3M and a velocity of 2v would have a momentum of 192 kg•m/s.

Answer: See Answers above.

Momentum is the product of mass and velocity. As such, the momentum of an object is directly proportional to the mass and directly proportional to the velocity. If the mass of an object is altered by some factor, then the momentum of the object is altered by that same factor. So if the mass is doubled, the momentum is doubled; the new value would be two times the original value. And if the mass is tripled, then the momentum is tripled; the new value would be three times the original value. If the velocity of an object is altered by some factor, then the momentum of the object is altered by that same factor. So if the velocity is doubled, the momentum is doubled; the new value would be two times the original value. And if the velocity is tripled, then the momentum is tripled; the new value would be three times the original value. This is shown in the work below.

54. An object with a mass M and a velocity v undergoes a collision and encounters a force of F for a time of t. The collision brings the object to a final rest position ...

... If the object had a mass of 2M and a velocity of v, then it would need an impulse which is 2F•t in order to be brought to rest.

... If the same object encountered a force of 2F, then it would bring it to rest in a time of 0.5t, The impulse would be the same size and the momentum change would be the same size.

... If the same object encountered a force of 10F, then it would bring it to rest in a time of 0.1t, The impulse would be the same size and the momentum change would be the same size.

... If the same object encountered a force of 0.2F, then it would bring it to rest in a time of 5t, The impulse would be the same (size and the momentum change would be the same size.

... If the object had a mass of 2M and a velocity of v encountered a force of 4F, then it would be brought to rest in a time of 0.5t. The impulse would be 2 times the original impulse and the momentum change would be 2 times the original impulse.

... If the object had a mass of 2M and a velocity of 2v encountered a force of 4F, then it would be brought to rest in a time of 1.0t. The impulse would be 4 times the original impulse and the momentum change would be 4 times the original impulse.

... If the object had a mass of 0.5M and a velocity of 4v encountered a force of 2F, then it would be brought to rest in a time of 1.0t. The impulse would be 2 times the original impulse and the momentum change would be 2 times the original impulse.

Answer: See Answers above.

In a collision, the impulse encountered by an object causes and is equal to the momentum change of the object. For objects being brought to rest, the change in momentum is simply equal to the original momentum. An object with twice the momentum would require twice the impulse to stop it. An object with one-half the momentum would require one-half the impulse to stop it. The momentum is the product of mass and velocity, so any alteration in the mass or the velocity or both would alter the required impulse by that same factor. The impulse results from a force acting over time and is the product of force and time. So twice the impulse can be achieved by the same force acting for twice the time or twice the force acting for the same time or even one-half the force acting for four times the time. These principles are illustrated below.

a. Two times the mass means two times the original momentum. Two times the impulse would be required to stop an object with two times the momentum.

b. If the same object with the same momentum (M•v) were stopped by twice the original force, then only one-half the original time would be required to stop it. Since the momentum change has not been altered in this case (an object with a momentum of M•v is still being stopped), the impulse is not altered either.

c. If the same object with the same momentum (M•v) were stopped by ten times the original force, then only one-tenth the original time would be required to stop it. Since the momentum change has not been altered in this case (an object with a momentum of M•v is still being stopped), the impulse is not altered either.

d. If the same object with the same momentum (M•) were stopped by one-fifth the original force, then five times the original time would be required to stop it. Since the momentum change has not been altered in this case (an object with a momentum of M•v is still being stopped), the impulse is not altered either.

e. If an object with twice the mass is being stopped, then twice the momentum change would occur. Two times the momentum change means that an impulse which is two times the original impulse would be required. To achieve two times the impulse, the product of force and time must be doubled. Since the force is four times bigger, the time needs to be one-half the original time.

f. If an object with twice the mass and twice the velocity is being stopped, then four times the momentum change would occur. Four times the momentum change means that an impulse which is four times the original impulse would be required. To achieve four times the impulse, the product of force and time must be quadrupled. Since the force is four times bigger, the time needs to be the same as the original time.

g. If an object with one-half the mass and four times the velocity is being stopped, then twice the momentum change would occur. Two times the momentum change means that an impulse which is two times the original impulse would be required. To achieve two times the impulse, the product of force and time must be doubled. Since the force is two times bigger, the time needs to be the same as the original time.

55. Two carts are placed next to each other on a low-friction track. The carts are equipped with a spring-loaded mechanism which allows them to impart an impulse to each other. Cart A has a mass of M and Cart B has a mass of M. The spring-loaded mechanism is engaged and then released. The impulse causes Cart A to be propelled forward with a velocity of 40 cm/s.

Cart B will be propelled backward with a velocity of 40 cm/s.

... If Cart B had a mass of 2M then it would be propelled backwards with a velocity of 20 cm/s.

... If Cart B had a mass of 0.5M then it would be propelled backwards with a velocity of 80 cm/s.

... If Cart B has a mass of 2M then it would be propelled backwards with a momentum which is the same as the original momentum.

... If Cart B has a mass of 2M then it would encounter an impulse which is the same as the original impulse.

Part E: Problem-Solving

57. A 0.530-kg basketball hits a wall head-on with a forward speed of 18.0 m/s. It rebounds with a speed of 13.5 m/s. The contact time is 0.100 seconds. (a) determine the impulse with the wall, (b) determine the force of the wall on the ball.

58. A 4.0-kg object has a forward momentum of 20. kg•m/s. A 60. N•s impulse acts upon it in the direction of motion for 5.0 seconds. A resistive force of 6.0 N then impedes its motion for 8.0 seconds. Determine the final velocity of the object.

59. A 3.0-kg object is moving forward with a speed of 6.0 m/s. The object then encounters a force of 2.5 N for 8.0 seconds in the direction of its motion. The object then collides head-on with a wall and heads in the opposite direction with a speed of 5.0 m/s. Determine the impulse delivered by the wall to the object.

61. A 2.0-kg box is attached by a string to a 5.0-kg box. A compressed spring is placed between them. The two boxes are initially at rest on a friction-free track. The string is cut and the spring applies an impulse to both boxes, setting them in motion. The 2.0-kg box is propelled backwards and moves 1.2 meters to the end of the track in 0.50 seconds. Determine the time it takes the 5.0-kg box to move 0.90 meters to the opposite end of the track.

62. Two children are playing with a large snowball while on ice skates on a frozen pond. The 33-kg child tosses the 5.0-kg snowball, imparting a horizontal speed of 5.0 m/s to it. The 33-kg child is 4.0 meters from a 28-kg child and 8.0 meters from the edge of the pond (located behind him). Assuming negligible friction, how much time elapses between when the 28-kg child gets hit by the snowball and when the 33-kg child reaches the edge of the pond?

63. A 2.8-kg physics cart is moving forward with a speed of 45 cm/s. A 1.9-kg brick is dropped from rest and lands on the cart. The cart and brick move together across the horizontal surface. Assume an isolated system.

64. In a physics lab, a 0.500-kg cart moving at 36.4 cm/s collides inelastically with a second cart which is initially at rest. The two carts move together with a speed of 21.8 cm/s after the collision. Determine the mass of the second cart.

65. A 9230-kg truck collides head on with a 1250-kg parked car. The vehicles entangle together and slide a linear distance of 10.6 meters before coming to rest. Assuming a uniform coefficient of friction of 0.820 between the road surface and the vehicles, determine the pre-collision speed of the truck.

66. A classic physics demonstration involves firing a bullet into a block of wood suspended by strings from the ceiling. The height to which the wood rises below its lowest position is mathematically related to the pre-collision speed of the bullet. If a 9.7-gram bullet is fired into the center of a 1.1-kg block of wood and it rises upward a distance of 33 cm, then what was the pre-collision speed of the bullet?

68. Two billiard balls, assumed to have identical mass, collide in a perfectly elastic collision. Ball A is heading East at 12 m/s. Ball B is moving West at 8.0 m/s. Determine the post-collision velocities of Ball A and Ball B.

69. A 1.72-kg block of soft wood is suspended by two strings from the ceiling. The wood is free to rotate in pendulum-like fashion when a force is exerted upon it. A 8.50-g bullet is fired into the wood. The bullet enters the wood at 431 m/s and exits the opposite side shortly thereafter. If the wood rises to a height of 13.8 cm, then what is the exit speed of the bullet?

70. In a physics lab, the pitching speed of a student is determined by throwing a baseball into a box and observing the box's motion after the catch. A measurement of the the distance the box slides across a rough surface of known coefficient of friction will allow one to determine the pre-impact speed of the pitched ball. If a 0.256-kg ball hits a 3.46-kg box and the ball and box slide a distance of 2.89 meters across a surface with a coefficient of friction of 0.419, then what is the pre-impact speed of the pitched ball?

71. Two ice skaters collide on the ice. A 39.6-kg skater moving South at 6.21 m/s collides with a 52.1-kg skater moving East at 4.33 m/s. The two skaters entangle and move together across the ice. Determine the magnitude and direction of their post-collision velocity.

72. In a physics lab, two carts collide elastically on a level, low-friction track. Cart A has a mass of 1.500 kg and is moving east at 36.5 cm/s. Cart B has a mass of 0.500 kg and is moving West at 42.8 cm/s. Determine the post-collision velocities of the two carts.